.. index:: proof, parallel

.. _I.27:
.. _parallel lines 1:

the angles produced from a line intersecting parallel lines are equal
=====================================================================

  I.27

  If a straight line falling on two straight lines [1]_ make the alternate
  angles [2]_ equal to one another, the straight lines will be parallel to one
  another.

  -- Euclid

.. image:: elem.1.prop.27.png
   :align: right
   :width: 300px

For let the straight line `EF` falling on the two straight lines `AB`, `CD` make the alternate angles `AEF`, `EFD` equal to one another;

I say that `AB` is parallel to `CD`.

For, if not, `AB`, `CD` when produced will meet either in the direction of `B`, `D` or towards `A`, `C`. [3]_

Let them be produced and meet, in the direction of `B`, `D`, at `G`.

Then, in the triangle `GEF`, the exterior angle `AEF` is equal to the interior and opposite angle `EFG`: which is impossible. [I.16]

Therefore `AB`, `CD` when produced will not meet in the direction of `B`, `D`.

Similarly it can be proved that neither will they meet towards `A`, `C`. 

But straight lines which do not meet in either direction are parallel; [I.def.23] 

- therefore `AB` is parallel to `CD`.

Therefore etc.

- Q. E. D.

## References

[I.def.23]: /elem.1.def.23 "Book 1 - Definition 23"
[I.16]: /elem.1.16 "Book 1 - Proposition 16"

## Footnotes

.. [1] falling on two straight lines,
    <foreign lang="greek">εὶς δύο εὐθείας ἐμπίπτουσα</foreign>, the phrase
   being the same as that used in <a href="/elem.1.post.5">Post. 5</a>, meaning
   a <em>transversal</em>.

.. [2] the alternate angles,
    <foreign lang="greek">αἱ ἐναλλὰξ γωνίαι</foreign>. Proclus (<xref n="Proc.
   p. 357, 9" from="ROOT" to="DITTO">p. 357, 9</xref>) explains that Euclid
   uses the word <em>alternate</em> (or, more exactly, <em>alternately</em>,
   <foreign lang="greek">ἐναλλάξ</foreign>) in two connexions, (1) of a certain
   transformation of a proportion, as in Book V. and the arithmetical Books,
   (2) as here, of certain of the angles formed by parallels with a straight
   line crossing them. <title>Alternate</title> angles are, according to Euclid
   as interpreted by Proclus, those which are not on the same side of the
   transversal, and are not adjacent, but are separated by the transversal,
   both being within the parallels but one <quote>above</quote> and the other
   <quote>below.</quote> The meaning is natural enough if we imagine the four
   internal angles to be taken in cyclic order and <em>alternate</em> angles to
   be any two of them not successive but separated by one angle of the four.

.. [3] in the direction of B, D or towards A, C,
    literally <quote>towards the <em>parts B</em>, `D` or towards `A`, `C`,</quote> <foreign lang="greek">ἐπὶ τὰ Β, Δ μέρη ἢ ἐπὶ τὰ Α Γ</foreign>.